Answer:
41
Step-by-step explanation:
BDA = DBC+CBA
84=5x+3 + 6x-7
Combine like terms
84 = 11x-4
Add 4 to each side
84+4= 11x -4+4
88 =11x
Divide each side by 11
88/11 = 11x/11
8 = x
We want angle ABC
ABC = 6x-7
=6*8 -7
=48-7
=41
Answer:
x =8
Step-by-step explanation:
To answer this question you must find the point at which ![g(x)\geq f(x)](https://tex.z-dn.net/?f=g%28x%29%5Cgeq%20f%28x%29)
So, we have:
![x^2 + 2x + 5 \geq 8x + 16](https://tex.z-dn.net/?f=x%5E2%20%2B%202x%20%2B%205%20%5Cgeq%208x%20%2B%2016)
![x^2 + 2x -8x + 5 -16\geq0\\\\x^2 -6x -11\geq 0](https://tex.z-dn.net/?f=x%5E2%20%2B%202x%20-8x%20%2B%205%20-16%5Cgeq0%5C%5C%5C%5Cx%5E2%20-6x%20-11%5Cgeq%200)
To solve the quadratic function we use the quadratic formula
±
![\frac{-b \± \sqrt{b^2- 4ac}}{2a}](https://tex.z-dn.net/?f=%5Cfrac%7B-b%20%5C%C2%B1%20%5Csqrt%7Bb%5E2-%204ac%7D%7D%7B2a%7D)
Where:
![a = 1\\b =-6\\c = -11](https://tex.z-dn.net/?f=a%20%3D%201%5C%5Cb%20%3D-6%5C%5Cc%20%3D%20-11)
Then:
![\frac{-(-6) \± \sqrt{(-6)^2- 4(1)(-11)}}{2(1)}\\\\x = 7.47\\x = -1.472](https://tex.z-dn.net/?f=%5Cfrac%7B-%28-6%29%20%5C%C2%B1%20%5Csqrt%7B%28-6%29%5E2-%204%281%29%28-11%29%7D%7D%7B2%281%29%7D%5C%5C%5C%5Cx%20%3D%207.47%5C%5Cx%20%3D%20-1.472)
The line cuts the parabola by 2 points, x = -1.472 and x = 7.47.
You can verify that between x = -1.472 and x = 7.47. the line is greater than the parabola, but from x = 7.47, the parabola is always greater than the graph of the line.
Therefore the point sought is:
x = 7.47≈ 8
For this case we have that by definition, the equation of a line of the slope-intersection form is given by:
![y = mx + b](https://tex.z-dn.net/?f=y%20%3D%20mx%20%2B%20b)
Where:
m: It is the slope of the line
b: It is the cut-off point with the y axis
We have the following equation:
![y = -7x + 6](https://tex.z-dn.net/?f=y%20%3D%20-7x%20%2B%206)
With slope ![m_ {1} = - 7](https://tex.z-dn.net/?f=m_%20%7B1%7D%20%3D%20-%207)
By definition, if two lines are parallel then their slopes are equal. Thus, a parallel line will be of the form:
![y = -7x + b](https://tex.z-dn.net/?f=y%20%3D%20-7x%20%2B%20b)
We replace the point (5,5) through which the line passes and find "b":
![5 = -7 (5) + b\\5 = -35 + b\\5 + 35 = b\\b = 40](https://tex.z-dn.net/?f=5%20%3D%20-7%20%285%29%20%2B%20b%5C%5C5%20%3D%20-35%20%2B%20b%5C%5C5%20%2B%2035%20%3D%20b%5C%5Cb%20%3D%2040)
Finally, the equation is:
![y = -7x + 40](https://tex.z-dn.net/?f=y%20%3D%20-7x%20%2B%2040)
On the other hand, if two lines are perpendicular then the product of their slopes is -1. Thus, the slope of a perpendicular line will be:
![m_ {2} = \frac {-1} {m_ {1}} = \frac {-1} {- 7} = \frac {1} {7}](https://tex.z-dn.net/?f=m_%20%7B2%7D%20%3D%20%5Cfrac%20%7B-1%7D%20%7Bm_%20%7B1%7D%7D%20%3D%20%5Cfrac%20%7B-1%7D%20%7B-%207%7D%20%3D%20%5Cfrac%20%7B1%7D%20%7B7%7D)
Thus, the equation is of the form:
![y = \frac {1} {7} x + b](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%20%7B1%7D%20%7B7%7D%20x%20%2B%20b)
We substitute the point (5, -5) through which the line passes and find "b":
![-5 = \frac {1} {7} (5) + b\\-5 = \frac {5} {7} + b\\-5- \frac {5} {7} = b\\b = \frac {-35-5} {7}\\b = \frac {-40} {7}\\b = - \frac {40} {7}\\](https://tex.z-dn.net/?f=-5%20%3D%20%5Cfrac%20%7B1%7D%20%7B7%7D%20%285%29%20%2B%20b%5C%5C-5%20%3D%20%5Cfrac%20%7B5%7D%20%7B7%7D%20%2B%20b%5C%5C-5-%20%5Cfrac%20%7B5%7D%20%7B7%7D%20%3D%20b%5C%5Cb%20%3D%20%5Cfrac%20%7B-35-5%7D%20%7B7%7D%5C%5Cb%20%3D%20%5Cfrac%20%7B-40%7D%20%7B7%7D%5C%5Cb%20%3D%20-%20%5Cfrac%20%7B40%7D%20%7B7%7D%5C%5C)
Finally, the equation is:
![y = \frac {1} {7} x- \frac {40} {7}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%20%7B1%7D%20%7B7%7D%20x-%20%5Cfrac%20%7B40%7D%20%7B7%7D)
Answer:![y = -7x + 40\\y = \frac {1} {7} x- \frac {40} {7}](https://tex.z-dn.net/?f=y%20%3D%20-7x%20%2B%2040%5C%5Cy%20%3D%20%5Cfrac%20%7B1%7D%20%7B7%7D%20x-%20%5Cfrac%20%7B40%7D%20%7B7%7D)
Answer:
50
Step-by-step explanation: